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August  2022, 5(3): 173-185. doi: 10.3934/mfc.2021026

On bivariate Jain operators

Ankara University, Faculty of Science, Department of Mathematics, Str. Dögol, 06100, Beşevler, Ankara, Turkey

* Corresponding author: Gülen Başcanbaz-Tunca

Received  June 2021 Revised  September 2021 Published  August 2022 Early access  October 2021

In this paper we deal with bivariate extension of Jain operators. Using elementary method, we show that these opearators are non-increasing in $ n $ when the attached function is convex. Moreover, we demonstrate that these operators preserve the properties of modulus of continuity. Finally, we present a Voronovskaja type theorem for the sequence of bivariate Jain operators.

Citation: Münüse Akçay, Gülen Başcanbaz-Tunca. On bivariate Jain operators. Mathematical Foundations of Computing, 2022, 5 (3) : 173-185. doi: 10.3934/mfc.2021026
References:
[1]

U. Abel and O. Agratini, Asymptotic behaviour of Jain operators, Numer Algor., 71 (2016), 553-565.  doi: 10.1007/s11075-015-0009-3.

[2]

O. Agratini, Approximation properties of a class of linear operators, Math. Meth. Appl. Sci., 36 (2013), 2353-2358.  doi: 10.1002/mma.2758.

[3]

O. Agratini, A stop over Jain operators and their generalizations, An. Univ. Vest Timiş. Ser. Mat.-Inform., 56 (2018), 28-42. doi: 10.2478/awutm-2018-0014.

[4]

C. Bardaro and I. Mantellini, On Pointwise approximation properties of multivariate Semi-discrete sampling type operators, Results Math., 72 (2017), 1449-1472.  doi: 10.1007/s00025-017-0667-7.

[5]

F. Cao, C. Ding and Z. Xu, On multivariate Baskakov operator, J. Math. Anal. Appl., 307 (2005), 274-291. doi: 10.1016/j.jmaa.2004.10.061.

[6]

N. Çetin and G. Başcanbaz-Tunca, Approximation by Jain-Schurer operators, Facta Univ. Ser. Math. Inform., 35 (2020), 1343-1356.  doi: 10.22190/fumi2005343c.

[7]

E. W. Cheney and A. Sharma, Bernstein power series, Can. J. Math., 16 (1964), 241-252.  doi: 10.4153/CJM-1964-023-1.

[8]

E. Deniz, Quantitative estimates for Jain-Kantorovich operators, Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat., 65 (2016), 121-132.  doi: 10.1501/Commua1_0000000764.

[9]

M. Dhamija and N. Deo, Jain–Durrmeyer operators associated with the inverse Pólya–Eggenberger distribution, Appl. Math. Comput., 286 (2016), 15-22.  doi: 10.1016/j.amc.2016.03.015.

[10]

O. DoğruR. N. Mohapatra and M. Örkcü, Approximation properties of genaralized Jain operators, Filomat, 30 (2016), 2359-2366.  doi: 10.2298/FIL1609359D.

[11]

A. Farcaş, An asymptotic formula for Jain's operators, Stud. Univ. Babeş-Bolyai Math., 57 (2012), 511–517.

[12]

A. Farcaş, An approximation property of the generalized Jain's operators of two variables, Math. Sci. Appl. E-Notes, 1 (2013), 158-164. 

[13]

V. GuptaR. P. Agarwal and D. K. Verma, Approximation for a new sequence of summation-integral type operators, Adv. Math. Sci. Appl., 23 (2013), 35-42. 

[14]

V. Gupta and G. C. Greubel, Moment estimations of new Szász–Mirakyan–Durrmeyer operators, Appl. Math. Comput., 271 (2015), 540-547.  doi: 10.1016/j.amc.2015.09.037.

[15]

G. C. Jain, Approximation of functions by a new class of linear operators, J. Austral. Math. Soc., 13 (1972), 271-276.  doi: 10.1017/S1446788700013689.

[16]

Z. Li, Bernstein polynomials and modulus of continuity, J. Approx. Theory, 102 (2000), 171-174.  doi: 10.1006/jath.1999.3374.

[17]

G. M. Mirakjan, Approximation of continuous functions with the aid of polynomials, Dokl. Akad. Nauk SSSR, 31 (1941), 201-205. 

[18]

A. Olgun, Some properties of the multivariate Szász operators, C. R. Acad. Bulg. Sci., 65 (2012), 139-146. 

[19]

A. OlgunF. Taşdelen and A. Erençin, A generalization of Jain's operators, Appl. Math. Comput., 266 (2015), 6-11.  doi: 10.1016/j.amc.2015.05.060.

[20]

M. A. Özarslan, Approximation properties of Jain-Stancu operators, Filomat, 30 (2016), 1081-1088.  doi: 10.2298/FIL1604081O.

[21]

L. Rempulska and M. Skorupka, On Szasz-Mirakyan operators of functions of two variables, Le Matematiche, 53 (1998), 51-60. 

[22]

D. D. Stancu and E. I. Stoica, On the use Abel-Jensen type combinatorial formulas for construction and investigation of some algebraic polynomial operators of approximation, Stud. Univ. Babeş -Bolyai Math., 54 (2009), 167–182.

[23]

O. Szász, Generalization of S. Bernstein's polynomials to the infinite interval, J. Res. Natl. Bur. Stand., 45 (1950), 239-245.  doi: 10.6028/jres.045.024.

[24]

S. Tarabie, On Jain-Beta linear operators, Appl. Math. Inf. Sci., 6 (2012), 213-216. 

[25]

S. Umar and Q. Razi, Approximation of function by a generalized Szász operators, Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat., 34 (1985), 45-52.  doi: 10.1501/Commua1_0000000240.

show all references

References:
[1]

U. Abel and O. Agratini, Asymptotic behaviour of Jain operators, Numer Algor., 71 (2016), 553-565.  doi: 10.1007/s11075-015-0009-3.

[2]

O. Agratini, Approximation properties of a class of linear operators, Math. Meth. Appl. Sci., 36 (2013), 2353-2358.  doi: 10.1002/mma.2758.

[3]

O. Agratini, A stop over Jain operators and their generalizations, An. Univ. Vest Timiş. Ser. Mat.-Inform., 56 (2018), 28-42. doi: 10.2478/awutm-2018-0014.

[4]

C. Bardaro and I. Mantellini, On Pointwise approximation properties of multivariate Semi-discrete sampling type operators, Results Math., 72 (2017), 1449-1472.  doi: 10.1007/s00025-017-0667-7.

[5]

F. Cao, C. Ding and Z. Xu, On multivariate Baskakov operator, J. Math. Anal. Appl., 307 (2005), 274-291. doi: 10.1016/j.jmaa.2004.10.061.

[6]

N. Çetin and G. Başcanbaz-Tunca, Approximation by Jain-Schurer operators, Facta Univ. Ser. Math. Inform., 35 (2020), 1343-1356.  doi: 10.22190/fumi2005343c.

[7]

E. W. Cheney and A. Sharma, Bernstein power series, Can. J. Math., 16 (1964), 241-252.  doi: 10.4153/CJM-1964-023-1.

[8]

E. Deniz, Quantitative estimates for Jain-Kantorovich operators, Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat., 65 (2016), 121-132.  doi: 10.1501/Commua1_0000000764.

[9]

M. Dhamija and N. Deo, Jain–Durrmeyer operators associated with the inverse Pólya–Eggenberger distribution, Appl. Math. Comput., 286 (2016), 15-22.  doi: 10.1016/j.amc.2016.03.015.

[10]

O. DoğruR. N. Mohapatra and M. Örkcü, Approximation properties of genaralized Jain operators, Filomat, 30 (2016), 2359-2366.  doi: 10.2298/FIL1609359D.

[11]

A. Farcaş, An asymptotic formula for Jain's operators, Stud. Univ. Babeş-Bolyai Math., 57 (2012), 511–517.

[12]

A. Farcaş, An approximation property of the generalized Jain's operators of two variables, Math. Sci. Appl. E-Notes, 1 (2013), 158-164. 

[13]

V. GuptaR. P. Agarwal and D. K. Verma, Approximation for a new sequence of summation-integral type operators, Adv. Math. Sci. Appl., 23 (2013), 35-42. 

[14]

V. Gupta and G. C. Greubel, Moment estimations of new Szász–Mirakyan–Durrmeyer operators, Appl. Math. Comput., 271 (2015), 540-547.  doi: 10.1016/j.amc.2015.09.037.

[15]

G. C. Jain, Approximation of functions by a new class of linear operators, J. Austral. Math. Soc., 13 (1972), 271-276.  doi: 10.1017/S1446788700013689.

[16]

Z. Li, Bernstein polynomials and modulus of continuity, J. Approx. Theory, 102 (2000), 171-174.  doi: 10.1006/jath.1999.3374.

[17]

G. M. Mirakjan, Approximation of continuous functions with the aid of polynomials, Dokl. Akad. Nauk SSSR, 31 (1941), 201-205. 

[18]

A. Olgun, Some properties of the multivariate Szász operators, C. R. Acad. Bulg. Sci., 65 (2012), 139-146. 

[19]

A. OlgunF. Taşdelen and A. Erençin, A generalization of Jain's operators, Appl. Math. Comput., 266 (2015), 6-11.  doi: 10.1016/j.amc.2015.05.060.

[20]

M. A. Özarslan, Approximation properties of Jain-Stancu operators, Filomat, 30 (2016), 1081-1088.  doi: 10.2298/FIL1604081O.

[21]

L. Rempulska and M. Skorupka, On Szasz-Mirakyan operators of functions of two variables, Le Matematiche, 53 (1998), 51-60. 

[22]

D. D. Stancu and E. I. Stoica, On the use Abel-Jensen type combinatorial formulas for construction and investigation of some algebraic polynomial operators of approximation, Stud. Univ. Babeş -Bolyai Math., 54 (2009), 167–182.

[23]

O. Szász, Generalization of S. Bernstein's polynomials to the infinite interval, J. Res. Natl. Bur. Stand., 45 (1950), 239-245.  doi: 10.6028/jres.045.024.

[24]

S. Tarabie, On Jain-Beta linear operators, Appl. Math. Inf. Sci., 6 (2012), 213-216. 

[25]

S. Umar and Q. Razi, Approximation of function by a generalized Szász operators, Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat., 34 (1985), 45-52.  doi: 10.1501/Commua1_0000000240.

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